Rokhlin Dimension for Flows

We study the topological variant of Rokhlin dimension for topological dynamical systems (X,{\alpha},Z^m) in the case where X is assumed to have finite covering dimension. Finite Rokhlin dimension in this sense is a property that implies finite Rokhlin dimension of the induced action on C*-algebraic level, as was discussed in a recent paper by Ilan Hirshberg, Wilhelm Winter and Joachim Zacharias. In particular, it implies under these conditions that the transformation group C*-algebra has finite nuclear dimension. Generalizing a result of Yonatan Gutman, we show that free Z^m-actions on finite dimensional spaces satisfy a strengthened version of the so-called marker property, which yields finite Rokhlin dimension for said actions.

We characterize subhomogeneity for twisted étale groupoid $\text{C}^{*}$-algebras and obtain an upper bound on their nuclear dimension. As an application, we remove the principality assumption in recent results on upper bounds on the nuclear dimension of a twisted étale groupoid $\text{C}^{*}$-algebra in terms of the dynamic asymptotic dimension of the groupoid and the covering dimension of its unit space. As a non-principal example, we show that the dynamic asymptotic dimension of any minimal (not necessarily free) action of the infinite dihedral group $D_{\infty }$ on an infinite compact Hausdorff space $X$ is always one. So if we further assume that $X$ is second-countable and has finite covering dimension, then $C(X)\rtimes _{r} D_{\infty }$ has finite nuclear dimension and is classifiable by its Elliott invariant.

Let X be a compact infinite metric space of finite covering dimension and α : X → X a minimal homeomorphism. We prove that the crossed product $${\mathcal{C}(X) \rtimes_\alpha \mathbb{Z}}$$ absorbs the Jiang–Su algebra tensorially and has finite nuclear dimension. As a consequence, these algebras are determined up to isomorphism by their graded ordered K-theory under the necessary condition that their projections separate traces. This result applies, in particular, to those crossed products arising from uniquely ergodic homeomorphisms.

Let μ∈ P2(Rd), where P2(Rd) denotes the space of square integrable probability measures, and consider a Borel-measurable function Φ:P2(Rd)→R. In this paper we develop an antithetic Monte Carlo estimator (A-MLMC) for Φ(μ), which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws μN=1N∑ i=1NδX i, where (a) (Xi)i=1N is a sequence of i.i.d. samples from μ or (b) (Xi)i=1N is a system of interacting particles (diffusions) corresponding to a McKean–Vlasov stochastic differential equation (McKV-SDE). Each case requires a separate analysis. For a mean-field particle system, we also consider the empirical law induced by its Euler discretisation which gives a fully implementable algorithm. As by-products of our analysis, we establish a dimension-independent rate of uniform strong propagation of chaos, as well as an L2 estimate of the antithetic difference for i.i.d. random variables corresponding to general functionals defined on the space of probability measures.

We introduce two properties of dynamical systems on Polish metric spaces: closeability and linkability. We show that they imply density of ergodic measures in the space of invariant probability measures and the existence of a generic point for every invariant measure. In the compact case, it follows from our conditions that the set of invariant measures is either a singleton of a measure concentrated on a periodic orbit or the Poulsen simplex. We provide examples showing that closability and linkability are independent properties. Our theory applies to systems with the periodic specification property, irreducible Markov chains over a countable alphabet, certain coded systems including $\unicode[STIX]{x1D6FD}$-shifts and $S$-gap shifts, $C^{1}$-generic diffeomorphisms of a compact manifold $M$ and certain geodesic flows of a complete connected negatively curved manifold.

A definition of Lévy processes with values in the space of probability measures was introduced by Shiga and Tanaka (Electronic J. Prob. 11 (2006)). It is shown that the Lévy process with values in the space of probability measures in law has a modification satisfying a certain condition. The modification is a Lévy process in the sense of Shiga and Tanaka. The Lévy-Itô decomposition of sample paths of the Lévy process satisfying the condition is derived.

In this short communication we prove that the subspace Pn,n−1(X)of all probability measures P(X), whose supports consist of exactly n points is an (n−1)-dimensional topological manifold. A number of subspaces of the space of all probability measures having infinite dimension in the sense of dim, which are manifolds, are identified. We also consider individual subsets of the infinite compact set X, on which the space of probability measures is homotopy dense in the entire space. Three theorems on the topological properties of manifolds—subspaces of homotopy dense probability measures in the space of probability measures with finite supports on a compactum—are formulated and proven, and special cases of finite and infinite compactums are considered.

Mayer control problem with probabilistic uncertainty on initial positions

We prove that if K is a compact space and the space P(K K) of regular probability measures on K K has countable tightness in its weak topology, then L1( ) is separable for every 2 P(K). It has been known that such a result is a consequence of Martin's axiom MA(!1). Our theorem has several consequences; in particular, it general- izes a theorem due to Bourgain and Todor£evi¢ on measures on Rosenthal compacta. 1. Introduction. The tightness of a topological space X, denoted by (X), is the least cardinal number such that for every A X and x2 A there is a set A0 A withjA0j (X) and such that x2 A0. Throughout, K stands for a compact Hausdor topological space. By

The existence of an inverse limit of an inverse system of (probability) measure spaces has been investigated since the very beginning of the modern probability theory. Results from Kolmogorov [11], Bochner [2], Choksi [6], Metivier [15], Bourbaki [4], Mallory and Sion [12] among others have paved the way of the deep understanding of this problem. All the above results, however, call for some topological concepts, or at least the ones which are closely related topological ones. In this paper we investigate purely measurable inverse systems of (probability) measure spaces, and give a sufficient condition for the existence of a unique inverse limit. An example for the considered purely measurable inverse systems of (probability) measure spaces is also given.

This paper establishes topological semi-conjugacy between a symbolic dynamical system (\Sigma_{A},\sigma_{A}) and the system (M(X),T_{M}) induced on the space of probability measures by a compact topological dynamical system (X, T ) . Some conditions on an original system (X,T) are obtained for the induced system (M(X),T_M) to have a subsystem topologically semi-conjugate to (\Sigma_{A},\sigma_{A}) , and some relations between an original system (X,T) and the induced system (M(X),T_{M}) concerning semi-conjugacy to (\Sigma_{A},\sigma_{A}) are given. By using these results, several criteria of Li–Yorke chaos and Devaney chaos for the induced system (M(X),T_{M}) are established. Also, a characterization of the \mathcal{F} -mixing dynamical system (X,T) with Furstenberg family \mathcal{F} is obtained by means of the system (M(X),T_{M}) , which gives an answer to the question posed by Fu and Xing [Chaos Solitons Fractals (2012), 439–443].

A Mayer optimal control problem on Wasserstein spaces over Riemannian manifolds

We consider conditional McKean–Vlasov stochastic differential equations (SDEs), as the ones arising in the large-system limit of mean field games and particle systems with mean field interactions when common noise is present. The conditional time-marginals of the solutions to these SDEs are governed by non-linear stochastic partial differential equations (SPDEs) of the second order, whereas their laws satisfy Fokker–Planck equations on the space of probability measures. Our paper establishes two superposition principles: The first asserts that any solution of the SPDE can be lifted to a solution of the conditional McKean–Vlasov SDE, and the second guarantees that any solution of the Fokker–Planck equation on the space of probability measures can be lifted to a solution of the SPDE. We use these results to obtain a mimicking theorem which shows that the conditional time-marginals of an Itô process can be emulated by those of a solution to a conditional McKean–Vlasov SDE with Markovian coefficients. This yields, in particular, a tool for converting open-loop controls into Markovian ones in the context of controlled McKean–Vlasov dynamics.

Partial derivative with respect to the measure and its application to general controlled mean-field systems

We study various partially ordered spaces of probability measures and we determine which of them are lattices. This has important consequences for optimization problems with stochastic dominance constraints. In particular we show that the space of probability measures on $\mathbb{R}$ is a lattice under most of the known partial orders, whereas the space of probability measures on $\mathbb{R}^d$ typically is not. Nevertheless, some subsets of this space, defined by imposing strong conditions on the dependence structure of the measures, are lattices.